Context. Upcoming weak lensing surveys like Euclid will provide an unprecedented opportunity to quantify the geometry and topology of the cosmic web, in particular in the vicinity of lensing clusters. Aims. Understanding the connectivity of the cosmic web with unbiased mass tracers like weak lensing is of prime importance to probe the underlying cosmology, seek dynamical signatures of dark matter, and quantify environmental effects on galaxy formation. Methods. Mock catalogues of galaxy clusters are extracted from the N-body PLUS simulation. For each cluster, the aperture multipolar moments of the convergence are calculated in two annuli (inside and outside the virial radius). By stacking their modulus, a statistical estimator is built to characterise the angular mass distribution around clusters. The moments are compared to predictions from perturbation theory and spherical collapse. Results. The main weakly chromatic excess of multipolar power on large scales is understood as arising from the contraction of the primordial cosmic web driven by the growing potential well of the cluster. In the inner region, the initial quadrupole prevails, while centring suppresses odd multipoles, especially m=1. Predictions for the signal amplitude as a function of the cluster-centric distance, mass and redshift are presented. The prospects of measuring this signal are estimated for current and future lensing data sets. Inside the virial radius, multipoles up to m=8-12 can be measured at >10 sigma with 10 000 clusters at z=0.3. In the outer regions, around 4 Rvir, detection is possible up to m=8 but the noise from intervening large-scale structure dominates. Conclusions. The Euclid mission should provide all the necessary information to study the cosmic evolution of the connectivity of the cosmic web around lensing clusters using multipolar moments [abridged].